Positivity and radial symmetry of solutions to some variational problems in RN ✩

We minimize functionals J(v1, . . . , vn) ≡ RN (1/p) n i=1 |∇vi |p − F |x|, v1, . . ., vn in (W1,p(RN))n, subject to integral constraints RN Gij (vi ) = cij (j = 1, . . . , ki, i = 1, . . . , n). We prove, under fairly weak conditions on the functions F, Gij , that smooth minimizers are radially symmetric and do not change sign. We also show generalizations of this result to other variational problems associated to degenerate elliptic systems. Our proofs are based on rearrangement arguments and the strong maximum principle.  2004 Published by Elsevier Inc